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Order of Differential Equation

The order of differential equation is the highest derivative of the dependent variable with respect to the independent variable. The order of a differential equation further helps to find the degree of the differential equation, and also to perform numerous calculations of differential equations.

Let us know more about how to find the order of a differential equation, the steps for order

1.	How To Find Order of Differential Equation?
2.	Order of Differential Equation
3.	Examples on Order of Differential Equation
4.	Practice Questions on Order of Differential Equation
5.	FAQs on Order of Differential Equation

How To Find Order of Differential Equation?

The order of the differential equation can be found by first identifying the derivatives in the given expression of the differential equation. The different derivatives in a differential equation are as follows.

First Derivative:dy/dx or y'
Second Derivative: d²y/dx², or y''
Third Derivative: d³y/dx³, or y'''
nth derivative: dⁿy/dxⁿ, or y^{''''.....n times}

Further, the highest derivative present in the differential equation defines the order of the differential equation. Similar to a polynomial equation in x, a differential equation has derivatives of the dependent variable with respect to derivatives in the independent variable.

Order of Differential Equations

The order of a differential equation is the highest order of the derivative appearing in the equation. Consider the following differential equations,

dy/dx = e^x, (d⁴y/dx⁴) + y = 0, (d³y/dx³)² + x²(d²y/dx²) + xdy/dx + 3= 0

In above differential equation examples, the highest derivative are of first, fourth and third order respectively.

First Order Differential Equation

You can see in the first example, it is a first-order differential equation that has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’

Second-Order Differential Equation

The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d²y/dx² = f”(x) = y”

Order of Differential Equation

Related Topics

The following topics help in a better understanding of the order of the differential equations.

Examples on Order of Differential Equation

Example 1: Find the order of the following differential equations.

(a). 3(d²y/dx²) + x(dy/dx)³ = 0

(b). (y''')²+ x²(y')³ - 2x + 11 = 0

(c). (dy/dx)² + (dy/dx) - Cos³x = 0

Solution:

(a). This differential equation is of second order, as the highest derivative is the second derivative.

(b). The differential equation is of third order, as the highest derivative is the third derivative.

(c). The differential equation is of first order, as the equation has only the first derivatives.

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Practice Questions on Order of Differential Equation

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FAQs on Order of Differential Equation

How Do You Find the Order of Differential Equation?

The order of a differential equation can be found by identifying the highest derivative which can be found fin the differential equation. Let us check for the order of the differential equation from the following examples of differential equations.

dy/dx = e^x, (d⁴y/dx⁴) + y = 0, (d³y/dx³)² + x²(d²y/dx²) + xdy/dx + 3= 0

In these differential equations, the differential equations are of first, fourth and third order respectively.

What Is the Order and Degree of Differential Equation?

The degree of the differential equation is the power of the highest ordered derivative present in the equation. To find the degree of the differential equation, we need to have a positive integer as the index of each derivative.

How Do You Know If the Differential Equation Is A First Order or Second Order Differential Equation?

The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are of the first order. It has only the first derivative dy/dx. And the equation which includes the second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d²y/dx² = f”(x) = y”

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